When the pseudo-norm \({}_{^*}\)  is a norm over the space of measures \(()\) , the kernel is said to be definite. When it is a norm only when restricted to measures of total mass 0, it is said to be conditionally definite.
This means that if the kernel is definite (such as the Gaussian kernel), then the associated MMD norm is “unbalanced” and can be used to compare measures with different total mass.
It is known to metrize the weak* topology when the kernel is universal, i.e. when the set of functions \(\lbrace x\mapsto k(x,y), y\in \rbrace \)  is dense in \(()\)  [1]}, [2]}.
On \(=^d\) , typical examples of universal kernels are the Gaussian and Laplacian kernels, of the form \(k(x,y) = e^{-{x-y}^p/s}\)  for respectively \(p=2\)  and \(p=1\) , and \(s>0\) .
In contrast, the energy distance kernel \(k(x,y) = -{x-y}_2\)  is universal but only conditionally positive, so it can only be used to compare probability distributions (it is a “balanced” norm).
